| import Mathlib | |
| import OAI.Probability.SKGap.Matrix.MatrixEntryCLM | |
| import OAI.Probability.SKGap.Gaussian.GaussianProductIntegral | |
| import OAI.Probability.SKGap.Localization.IntegralProductDeviation | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set | |
| open scoped NNReal ENNReal | |
| open Filter | |
| open scoped Topology | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius NNReal | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Real Set Filter | |
| open MeasureTheory.Measure | |
| open scoped ENNReal NNReal MeasureTheory Topology | |
| open MeasureTheory | |
| noncomputable section | |
| namespace SKGap | |
| open Matrix Real MeasureTheory ProbabilityTheory | |
| open scoped BigOperators Matrix.Norms.Frobenius NNReal ENNReal | |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| theorem goe_inverse_loop_centered {r : ℝ} (hr : 0 ≤ r) (B : Matrix ι ι ℝ) (i : ι) | |
| {K : Matrix ι ι ℝ → Matrix ι ι ℝ} {L T : ℝ≥0} {C : ℝ} | |
| {s : Set (MatrixCoordinates ι → ℝ)} | |
| (hK : LipschitzWith L K) (hC : ∀ M, opNorm (K M) ≤ C) | |
| (hT : LipschitzWith T (fun g : EuclideanSpace ℝ (MatrixCoordinates ι) => | |
| Matrix.trace (K (goeMatrix r g)*B))) (hT0 : 0 < T) | |
| (hs : MeasurableSet s) | |
| (hder : ∀ g ∈ s, ∀ b, HasDerivAt | |
| (fun t : ℝ => K (goeMatrix r g+t • symmetricElementary i b) b i) | |
| ((K (goeMatrix r g)*B*symmetricElementary i b*K (goeMatrix r g)) b i) 0) : | |
| let μ := Measure.pi (fun _ : MatrixCoordinates ι => gaussianReal 0 1) | |
| let Lπ : ℝ≥0 := L*(⟨sqrt (2*r),sqrt_nonneg _⟩ * | |
| (Fintype.card (MatrixCoordinates ι) : ℝ≥0) ^ (1/(2:ℝ≥0∞)).toReal) | |
| |(∫ g, (goeMatrix r g*K (goeMatrix r g)) i i ∂μ)- | |
| r*(∫ g, Matrix.trace (K (goeMatrix r g)*B) ∂μ)*(∫ g, K (goeMatrix r g) i i ∂μ)| ≤ | |
| (Fintype.card ι:ℝ)*(sqrt (2*r)*(Lπ:ℝ)+r*(2*C*opNorm B*C))*μ.real sᶜ+ | |
| r*(C*((2*exp (π^2/8))*(T:ℝ))+C*opNorm B*C) := by | |
| intro μ Lπ | |
| have hC0 : 0 ≤ C := (norm_nonneg _).trans (hC 0) | |
| have hcomp : LipschitzWith Lπ (fun g => K (goeMatrix r g)) := | |
| hK.comp (goeMatrix_pi_lipschitz r) | |
| have hf (a b : ι) : LipschitzWith Lπ (fun g => K (goeMatrix r g) a b) := by | |
| simpa only [one_mul] using! (matrixEntry_lipschitz a b).comp hcomp | |
| have hc (g : MatrixCoordinates ι → ℝ) (a b : ι) : | |
| ‖K (goeMatrix r g) a b‖ ≤ C := by | |
| simpa only [Real.norm_eq_abs] using (matrix_entry_le_opNorm _ a b).trans (hC _) | |
| have hd (g : MatrixCoordinates ι → ℝ) (b : ι) : | |
| ‖(K (goeMatrix r g)*B*symmetricElementary i b*K (goeMatrix r g)) b i‖ ≤ | |
| 2*C*opNorm B*C := by | |
| have he : ‖(K (goeMatrix r g)*B*symmetricElementary i b*K (goeMatrix r g)) b i‖ ≤ | |
| ‖K (goeMatrix r g)*B*symmetricElementary i b*K (goeMatrix r g)‖ := by | |
| simpa only [Real.norm_eq_abs] using! | |
| matrix_entry_le_frobenius (K (goeMatrix r g)*B*symmetricElementary i b*K (goeMatrix r g)) b i | |
| apply he.trans | |
| apply (matrix_inverse_diff_bound _ _ _).trans | |
| calc | |
| _ ≤ C*(opNorm B*2)*C := by | |
| exact mul_le_mul | |
| (mul_le_mul (hC _) (mul_le_mul_of_nonneg_left (symmetricElementary_norm_le i b) (norm_nonneg _)) | |
| (mul_nonneg (norm_nonneg _) (norm_nonneg _)) hC0) | |
| (hC _) (norm_nonneg _) (mul_nonneg hC0 (mul_nonneg (norm_nonneg _) (by norm_num))) | |
| _ = _ := by ring | |
| have hz (g : MatrixCoordinates ι → ℝ) : | |
| ‖∑ b, (K (goeMatrix r g)*B) b i*K (goeMatrix r g) b i‖ ≤ C*opNorm B*C := by | |
| rw [Real.norm_eq_abs] | |
| apply (goe_cross_contraction_bound _ _ _).trans | |
| exact mul_le_mul (mul_le_mul_of_nonneg_right (hC _) (norm_nonneg _)) (hC _) | |
| (norm_nonneg _) (mul_nonneg hC0 (norm_nonneg _)) | |
| have hzi : Integrable (fun g => ∑ b, (K (goeMatrix r g)*B) b i*K (goeMatrix r g) b i) μ := | |
| Integrable.mono' (integrable_const (C*opNorm B*C)) (by | |
| have hc := hcomp.continuous | |
| exact (continuous_finsetSum _ (fun b _ => | |
| ((hc.mul continuous_const).matrix_elem b i).mul (hc.matrix_elem b i))).aestronglyMeasurable) | |
| (ae_of_all _ hz) | |
| have hi : Integrable (fun g => Matrix.trace (K (goeMatrix r g)*B)) μ := | |
| gaussianProduct_integrable_lipschitz hT | |
| have ht : (∫ g, |Matrix.trace (K (goeMatrix r g)*B)- | |
| ∫ h, Matrix.trace (K (goeMatrix r h)*B) ∂μ| ∂μ) ≤ (2*exp (π^2/8))*(T:ℝ) := | |
| gaussianProduct_centered_L1 hT hT0 | |
| exact integral_loop_deviation hr hi (hf i i).continuous.aestronglyMeasurable | |
| (ae_of_all _ (fun g => hc g i i)) hzi (ae_of_all _ hz) ht | |
| (goe_inverse_loop_raw hr B i hf hc hd hs hder) | |
| end SKGap | |
| noncomputable section | |
| open MeasureTheory Set NormedSpace | |
| open scoped Topology | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |